TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 10, October 2008, Pages 5229–5246 S 0002-9947(08)04603-5 Article electronically published on May 20, 2008

C0-COARSE OF COMPLEMENTS OF Z-SETS IN THE HILBERT CUBE

E. CUCHILLO-IBA´ NEZ,˜ J. DYDAK, A. KOYAMA, AND M. A. MORON´

Abstract. Motivated by the Chapman Complement Theorem, we construct an isomorphism between the topological category of compact Z-sets in the Hilbert cube Q and the C0-coarse category of their complements. The C0- coarse morphisms are, in this particular case, intrinsically related to uniformly continuous proper maps. Using that fact we are able to relate in a natu- ral way some of the topological invariants of Z-sets to the geometry of their complements.

1. Introduction In 1972 T. Chapman [3] published what many consider the deepest and most beautiful results in shape theory. The one that stands out is the following Comple- ment Theorem in Shape Theory: Two compact Z-sets in the Hilbert cube have the same shape if and only if their complements are homeomorphic. More generally, he constructed an isomorphism of categories allowing one to describe Borsuk’s Shape Theory in terms of proper maps of complements of Z-sets in the Hilbert cube. Combining both results Chapman proved a rigidity theorem: The complements of two compact Z-sets (or two contractible Q-manifolds admit- ting a boundary [5]) in the Hilbert cube have the same weak proper homotopy type if and only if those complements are homeomorphic. Recently [1] suggested a possibility to describe topological invariants of compact Z-sets of the Hilbert cube in terms of -uniform invariants of their comple- ments. In particular two questions were posed there. They were related to the canonical copy of a compact inside its hyperspace with the Hausdorff metric. When the is a non-degenerate Peano continuum those ques-

Received by the editors July 17, 2006. 2000 Subject Classification. Primary 18B30, 54D35, 54E15; Secondary 54C55, 54E35, 54F45. Key words and phrases. Covering dimension, asymptotic dimension, C0-coarse structure, ANR-space, C0-coarse morphism, uniformly continuous map, compact Z-, Higson-Roe com- pactification and corona. The first and fourth named authors were supported by the MEC, MTM2006-0825.

c 2008 American Mathematical Society Reverts to public domain 28 years from publication 5229

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tions are about Z-copies in a Hilbert cube. When translated to our current context the questions become: 1. How can the covering dimension of a compact Z-set be described from the outside? 2. What is a manifold from the outside? The above paragraphs provide the main motivation for this paper. Following the spirit of the isomorphism of categories constructed by Chapman in [3] we prove that the topological category for compact Z-sets of the Hilbert cube is isomorphic to the C0-coarse geometry category of their complements. The definition of this C0-coarse geometry category requires a fixed metric on the Hilbert cube, but soon it is pointed out that the definition is, in fact, independent of the chosen (equivalent) metric. The C0-coarse structure of a metric space was introduced by Wright [16] (see also [17]) for different purposes from those pursued herein. We recommend [14] for the basic definitions of coarse spaces. Nevertheless we will recall some of these definitions, if needed. The objects in this category are the complements of Z-sets with the C0-coarse structure induced by a metric in the Hilbert cube. The morphisms are the equiv- alence classes of close coarse maps between them. See [16], [17] and [14] for the basic definitions of C0-coarse maps and C0-close maps. We also obtain our rigidity result. In fact, after the construction of the isomorphism of categories, it is very easy to see that the complements of two compact Z-sets are C0-coarse equivalent if and only if those complements are uniformly homeomorphic (with respect to the metrics induced by the one fixed on the Hilbert cube). In view of the equivalence of categories described above it is natural to consider the problem of describing topological invariants of Z-sets in terms of C0-coarse invariants of their complements. That is how the asymptotic dimension of such a coarse structure on the complement appears as the adequate tool to describe, from the outside, the topological dimension of a Z-set. To do it efficiently we use some results of [7]. In fact we prove that the Hilbert cube is the Higson-Roe com- pactification of the complement of a Z-set for this coarse structure. Moreover, its Higson-Roe corona is exactly that Z-set. That fact allows us to show that the di- mension can be interpreted in terms of the asymptotic dimension of such a coarse space. See [14] and [7] for the definitions of asymptotic dimension, Higson-Roe compactification, and corona for arbitrary coarse categories. So, in some sense, we reinforce the relation between the topological and coarse definitions of dimension and, in particular, we solve the first question of [1]. We also reinterpret the con- cept of external dimension introduced by Mrozik [11] as a characterization of the C0-asymptotic dimension of the complement of a Z-set analogous to that of the topological dimension using extensions of maps to spheres. The concept of asymptotic dimension was defined by Gromov [8] to study as- ymptotic invariants of discrete groups, mainly the fundamental group of manifolds. For finitely generated groups Gromov used the bounded coarse geometry associated to the word metric defined by means of a finite set of generators. The main fact in [8] was that such coarse geometry does not depend on the chosen generating set. Motivated by this, some authors (see for example Dranishnikov [6], Roe [12, 13] and references there) began to develop the asymptotic or large-scale ge- ometry. Our paper is related to [6] as we also connect topological invariants of compact spaces to coarse concepts (in our case to C0-coarse concepts). In fact,

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due to the isomorphism of categories we construct, this paper could help in the program of finding suitable translations of topological concepts to coarse geometry as pursued in [6], because it could be possible to relate all the topological invariants between compacta to the C0-coarse structure of their complements, by means of the categorical isomorphism, and then to adapt them to the general definition of coarse geometry. We do not follow this line herein. On the contrary, due to the fact that, in our case, the coarse morphisms can be represented by proper uniformly continuous functions, we stay as close as possible to Topology in the procedure of translating topological invariants of compact Z-sets to C0-coarse invariants of their complements. In [14], Roe gives a definition of asymptotic dimension for general coarse spaces. Later on Grave [7] proposed other definitions (translating some characterizations from the topological framework). All of them are equivalent in the realm of proper coarse spaces. We complete the paper by reinterpreting the ANR-property (a necessary prop- erty for being a manifold; see the second question above) and the group of auto- homeomorphisms in terms of the C0-coarse structure. We also take this opportunity to describe the homotopical category of compact metric spaces in terms of proper and uniformly continuous maps between their complements. The authors are very grateful to the referee whose detailed suggestions improved the exposition of the paper significantly.

2. Equivalence of categories and applications In this section we establish that the topological category of compacta (the most geometric topological spaces, namely compact and metrizable) can be described in terms of coarse geometry. More precisely, we need to use the so-called C0-coarse geometry introduced in [15] (see also [16]). Later on we translate some topological invariants to the coarse geometry language. As mentioned above we follow [14] for basic definitions and notation in Coarse Geometry. Recall (see [15] or [14, page 22]) that for each metric space (X, d)theC0-coarse structure is the collection E0 of all E ⊂ X × X for which the distance d, restricted to E, tends to zero at infinity; that is to say, E is controlled if for any ε>0 there is a compact K ⊂ X such that d(x, x) <εwhenever (x, x) ∈ E \ K × K. Consider the Hilbert cube (Q, d) with a fixed metric d reflecting its topology. Suppose that X ⊂ Q is a compact Z-set of Q and denote by (Q \ X, E0) the coarse space Q \ X with the C0-coarse structure associated to the metric d |(Q\X)×(Q\X). It is obvious that if X = ∅ is a closed Z-set in Q, then the metric d |(Q\X)×(Q\X) is never a proper metric (recall that a metric is proper when the compact are precisely the closed and bounded sets). On the other hand, there is a concept of proper coarse structure ([14, page 25]) which is important in coarse geometry. To establish that notion first let us recall some definitions from [14]. A coarse structure E on X is a family of subsets E (called controlled sets)of X × X satisfying the following properties:

(1) The diagonal ∆ = {(x, x)}x∈X belongs to E. (2) E1 ∈E implies E2 ∈E for every E2 ⊂ E1.

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−1 −1 (3) E ∈E implies E ∈E,whereE = {(y, x)}(x,y)∈E . (4) E1,E2 ∈E implies E1 ∪ E2 ∈E. (5) E,F ∈E implies E ◦ F ∈E,whereE ◦ F consists of (x, y) such that there is z ∈ X so that (x, z) ∈ E and (z,y) ∈ F .

A subset B ⊂ X is bounded for E (C0-bounded in the case of E0)ifB × B ∈E (see Proposition 2.16 in [14] for equivalent definitions of boundedness). For every E ⊂ X × X and K ⊂ X let E[K]={x ∈ X |∃x ∈ K, (x,x) ∈ E}, E−1[K]={x ∈ X |∃x ∈ K, (x, x) ∈ E}. When X is a , E ⊂ X × X is said to be proper (see Definition 2.1 in [14]) if E[K]andE−1[K] are relatively compact whenever K ⊂ X is relatively compact. Given a paracompact Hausdorff topological space X, a coarse structure on X is proper (see Definition 2.22 in [14]) if:

(i) There is a controlled neighborhood of the diagonal ∆X ,and (ii) Every (in the coarse sense) is relatively compact.

N. Wright [15] proved that the C0-coarse structure of a proper metric space (Y,d) is proper. For the sake of completeness, we are going to prove that the C0-coarse structure of (Q \ X, d |(Q\X)×(Q\X))isproperinspiteofd |(Q\X)×(Q\X) not being proper (see exercise 2.25 in [14]).

Proposition 1. Let (Q, d) be the Hilbert cube with a fixed metric d.TheC0-coarse structure E0 of Q \ X is a proper coarse structure for any compact Z-set X ⊂ Q.

Proof. For any point x ∈ Q \ X put εx = d(x, X) > 0. Let ε ε W = {(x, y) ∈ (Q \ X) × (Q \ X) | d(x, y) < x and d(x, y) < y }. 3 3

It is clear that the diagonal ∆Q\X is contained in W . We are going to prove that given (x, x) ∈ ∆Q\X there exists n ∈ N such that ε ε A = B(x, x ) × B(x, x ) (x,x) n n ∈ εx εx is contained in W .Infact,if(α, β) A(x,x),thend(α, x) < n and d(β,x) < n . Consider zα ∈ X with d(α, zα)=d(α, X)=εα.Then ε ε = d(x, X) ≤ d(x, z ) ≤ d(x, α)+d(α, z ) < x + ε . x α α n α Consequently ε n − 1 ε − x = ε <ε . x n n x α Hence ε ε 2 2 n 2 d(α, β) ≤ d(α, x)+d(x, β) < x + x = ε < ε = ε . n n n x n n − 1 α n − 1 α Analogously 2 d(α, β) < ε . n − 1 β Now it is sufficient to consider n = 7 in order to obtain ε ε d(α, β) < α and d(α, β) < β , 3 3

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which implies that (α, β) ∈ W and, in this manner, we have shown that W is a neighborhood of ∆Q\X . In order to establish that W is C0-controlled, for each δ>0, take the compact  \ ∈ \ × { εx εx } subset K = Q B(X, δ). If (x, x ) W K K,thend(x, x ) < min 3 , 3 . Since (x, x ) ∈/ K × K we can suppose that x/∈ K.Soεx = d(x, X) <δand ε δ d(x, x) < x < <δ. 3 3

Therefore W is C0-controlled. Let E ⊂ (Q \ X) × (Q \ X)beaC0-controlled set and K ⊂ Q \ X be a relatively compact set in Q \ X. The subset E[K]={x ∈ Q \ X |∃x ∈ K, (x,x) ∈ E} is relatively compact in Q \ X. Indeed, suppose that K is compact and E[K]is not relatively compact in Q \ X. In this case there exist a point x0 ∈ X and a sequence {xn}n∈N ⊂ E[K] with lim xn = x0 in Q. Fix a sequence {kn}n∈N ⊂ K n→∞ with (xn,kn) ∈ E for all n ∈ N. From the compactness of K we assume that lim kn = k0 ∈ K. Consequently, lim (xn,kn)=(x0,k0), but d(x0,k0) > 0; thus n→∞ n→∞ the distance function d, restricted to E, doesn’t tend to zero at infinity and this contradicts that E is a C0-controlled set. When K ⊂ Q \ X is relatively compact, since E[K] ⊂ E[ClQ\X(K)], E[K] is also relatively compact. −1 −1 For every E ∈E0 we have E ∈E0,soE [K] is relatively compact for each subset K relatively compact in Q \ X. Thus we proved that any E ∈E0 is proper. Finally, take a bounded subset (in the C0-sense) B ⊂ Q \ X. From Proposition 2.16 in [14], B = E[{p}]forsomeE ∈E0 and certain p ∈ Q \ X.SinceE is proper it follows that B is relatively compact and the proof is completed. 

Some of the main concepts in coarse geometry are those of a coarse map and close coarse maps. We are going to recall herein the versions for the C0-coarse geometry. See [14, pages 23-24] for the general concepts. Let (Q, d) be the Hilbert cube, with a fixed metric d,andX, Y be compact Z-sets in Q. A function f : Q \ X −→ Q \ Y (not necessarily continuous) is said to be a C0-coarse map if the following conditions are satisfied: −1 a) f (B)isC0-bounded in Q \ X if B is C0-bounded in Q \ Y . b) If E ⊂ (Q\X)×(Q\X)isC0-controlled, then (f ×f)(E)={(f(x),f(y)) : (x, y) ∈ E} is C0-controlled in Q \ Y .

Recall also that, in our context, being C0-bounded is equivalent to being rela- tively compact. Two C0-coarse maps, f,g : Q \ X −→ Q \ Y ,areC0-close if the subset {(f(x),g(x)) : x ∈ Q \ X} is C0-controlled in Q \ Y . In order to establish an isomorphism of categories we need to introduce some notation. Given the Hilbert cube (Q, d) with a fixed metric, denote by Z the category whose objects Ob(Z) are compact Z-sets of Q and the morphisms are continuous functions between Z-sets. Denote also by C0(Z) the category whose objects are the complements, in Q, of objects in Z with the C0-coarse structure induced by the restriction of the metric d. The morphisms between objects in C0(Z) are the equivalence classes, under the relation of C0-closeness, of C0-coarse maps. Note that, from the second paragraph in [14, page 25], the composition of classes

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is well defined by means of the class of the composition of arbitrary corresponding representations. Theorem 2. There is an isomorphism of categories

T : C0(Z) −→ Z so that the action on objects is defined by T (Q \ X, E0) = X. Proof. Define T on objects as stated above. Consider a C0-coarse map f : Q \ X −→ Q \ Y .SinceX is a Z-set, and therefore nowhere dense in Q,givenx0 ∈ X there exists a sequence {αn}n∈N ⊂ Q \ X with lim αn = x0. Obviously the set E = {(αn,αn+1): n ∈ N} is C0-controlled in n→∞ Q \ X.Consequently(f × f)(E)={(f(αn),f(αn+1)) : n ∈ N} is C0-controlled in Q \ Y .Letβ ∈ Q be an accumulation point of the sequence {f(αn)}n∈N and { } { } choose a subsequence αnk k∈N of αn n∈N such that limk f(αnk )=β. Assume ∈ \ { } ∪{ } that β Q Y . In that case, the compact K = f(αnk ) k∈N β is a subset of −1 Q \ Y .Asf is a C0-coarse map, f (K) should be a relatively compact subset in Q\X, but this is impossible because f −1(K) contains a sequence converging to the point x0 ∈ X. Thus we have proved that any accumulation point of {f(αn)}n∈N must be in Y . Consider β1,β2 ∈ Y to be two accumulation points of {f(αn)}n∈N and consider { } { } { } two subsequences αnk k∈N, αmk k∈N of αn n∈N such that limk f(αnk )=β1 and { } limk f(αmk )=β2. Take the sequence γj j∈N defined by ∈ N γ2l = αnl ,γ2l−1 = αml for l .

Clearly lim γj = x0 ∈ X, {γj}j∈N ⊂ Q \ X,thesetL = {(γj,γj+1): j ∈ N} is j→∞ C0-controlled in Q \ X and (f × f)(L)isC0-controlled in Q \ Y . Hence when we fix a number δ>0thereexistsn ∈ N such that if (x, y) ∈ (f × f)(L) \ (Kn × Kn), then d(x, y) <δ,whereKn = Q \ B(Y,1/n)=Q \{x ∈ Q ; d(x, Y ) < 1/n}.Bythe construction of the sequence {γj}j∈N it is obvious that, for each natural number n ∈ N,thereexistsjn ∈ N such that f(γj) ∈ B(Y,1/n) for any j ≥ jn.Sofor every δ>0thereisanumberjδ ∈ N such that d(f(γj),f(γj+1)) <δfor j ≥ jδ and consequently d(β1,β2) = 0, that is, β1 = β2. In this manner, the sequence { } f(αn) n∈N has a unique accumulation point, denote it by yx0 , that belongs to Y . Due to the compactness of Q, this means that, in fact, lim f(αn)=yx . n→∞ 0 ∈ The point yx0 Y does not depend on the choice of the initial sequence because if {α }n∈N ⊂ Q\X is another sequence, with lim α = x0, applying the procedure n n→∞ n described above to the new sequence {σn}n∈N defined as ∈ N σ2l = αl ,σ2l−1 = αl for l , then necessarily, lim f(αn)=yx0 . n→∞ Through the assignment x → yx we obtain a map z(f):X −→ Y.

This is a continuous map. In fact, if lim xn = x0,with{xn}n∈N ∪{x0}⊂X, n→∞ for each n ∈ N consider a sequence {αn,k}k∈N such that limk αn,k = xn and {αn,k}k∈N ⊂ Q \ X. Using now a diagonal process one can construct a sequence of { } the form αn,kn n∈N such that d(αn,kn ,xn) < 1/n and d(f(αn,kn ),z(f)(xn)) < 1/n.

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Clearly {αn,k }n∈N is a convergent sequence and lim αn,k = x0. Moreover, given n n→∞ n ε>0, by the definition of z(f), there is n0 ∈ N such that ≥ d(f(αn,kn ),z(f)(x0)) <ε/2andd(f(αn,kn ),z(f)(xn)) <ε/2 for all n n0 and therefore d(z(f)(xn),z(f)(x0)) <ε for all n ≥ n0. Thus the function z(f) is continuous. In order to define T on the morphisms in C0(Z), we need to state that for two C0- close maps f and g, the equality z(f)=z(g) holds. Suppose that the coarse maps f,g : Q \ X −→ Q \ Y are C0-close, that is to say, E = {(f(x),g(x)) : x ∈ Q \ X} is C0-controlled in Q \ Y .Takeapointx0 ∈ X and a sequence {αn}n∈N ⊂ Q \ X such that lim αn = x0.Sincez(f)(x0) = lim f(αn) ∈ Y and z(g)(x0) = lim g(αn) ∈ n→∞ n→∞ n→∞ Y we deduce that for any compact K ⊂ Q \ Y there is a number n(K) ∈ N such that (f(αn),g(αn)) ∈ E \ (K × K)forn ≥ n(K). So, from the C0-controllability of the set E, it follows that z(f)(x0)=z(g)(x0). For any C0-coarse map f let us denote by [f]theC0-close class of f.Nowwe define the functor on the morphisms as T ([f]) = z(f).

T is faithful: Assume that T ([f]) = T ([g]) and consider representative C0-coarse maps f,g : Q \ X −→ Q \ Y . Suppose now that f and g are not C0-close. In that case, as E = {(f(x),g(x)) : x ∈ Q \ X} is not C0-controlled, there is a number ε0 > 0 such that for each compact Kn = Q \ B(Y,1/n), n ∈ N,there exists (λn,ρn)=(f(αn),g(αn)) ∈ E \ (Kn × Kn)withd(λn,ρn) ≥ ε0. Refining the process if necessary, we assume that λn = f(αn) ∈/ Kn for all n ∈ N.Byits definition, roughly speaking, a coarse map transforms controlled sets into controlled sets; furthermore, a C0-coarse map also transforms relatively compact subsets (the C0-bounded ones) into relatively compact subsets. Therefore, as the accumulation points of {λn}n∈N lie in Y , necessarily the accumulation points of {αn}n∈N belong to X. In this manner, without loss of generality, we can assume that lim αn = x0 n→∞ for some element x0 ∈ X.Thus

lim f(αn)=z(f)(x0)=T ([f])(x0)=T ([g])(x0)=z(g)(x0) = lim g(αn), n→∞ n→∞

and this equality contradicts the fact that d(f(αn),g(αn)) ≥ ε0 > 0 for all n ∈ N. So we conclude that the set E = {(f(x),g(x)) : x ∈ Q \ X} is C0-controlled in Q \ Y and hence [f]=[g]. T is full: Given two compact Z-sets of Q, X, Y ∈ OB(Z), and a f : X −→ Y , consider a continuous extension F : Q −→ Q of f.Since Y is a Z-set there exists a homotopy, see [3] or [4], H : Q × IQ −→ Q,where IQ =[0, diam(Q)], with H0 ≡ idQ and Ht(Q) ∩ Y = ∅ for any t ∈ IQ \{0}.The function f˜ : Q −→ Q x → f˜(x)=H(F (x),d(x, X)) is continuous and satisfies ˜ ˜ f |X ≡ f and f(Q \ X) ⊂ Q \ Y. So the map ˆ ˜ f ≡ f |Q\X :(Q \ X, d) −→ (Q \ Y,d)

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is a proper (in the topological sense) uniformly continuous function because fˆ is the restriction of the uniformly continuous function f˜. Moreover, for any compact set K ⊂ Q \ Y , fˆ−1(K)=f˜−1(K) is then compact. In order to establish that ˆ ˆ ˆ f is a C0-coarse map, we only need to state that (f × f)(E)isaC0-controlled set in (Q \ Y ) × (Q \ Y ) for any C0-controlled set E ⊂ (Q \ X) × (Q \ X). In- deed, for each ε>0, from the uniform continuity of fˆ there is δ>0 such that ˆ ˆ d(f(x), f(x )) <εwhenever x, x ∈ Q \ X with d(x, x ) <δ.AsE is C0-controlled it is possible to choose a compact subset Kδ ⊂ (Q \ X) × (Q \ X) with the property ˆ ˆ that if (x, x ) ∈ E \ (Kδ × Kδ), then d(x, x ) <δ. For every pair (f(x), f(x )) ∈ ˆ ˆ ˆ ˆ (f × f)(E) \ (f(Kδ) × f(Kδ)), we have (x, x ) ∈ E \ (Kδ × Kδ); hence d(x, x ) <δ ˆ ˆ ˆ and consequently d(f(x), f(x )) <ε. This shows that f is a C0-coarse map and it is obvious that T ([fˆ]) = f. Functorial behavior: T ([g] ◦ [f]) = T ([g]) ◦ T ([f]). Let f :(Q \ X, E0) −→ (Q \ Y,E0)andg :(Q \ Y,E0) −→ (Q \ Z, E0)betwoC0-coarse maps for X, Y, Z ∈Z. Take a sequence {xn}n∈N ⊂ Q \ X with lim xn = x0 ∈ X.Then n→∞ T ([f])(x0) = lim f(xn)and n→∞

T ([g] ◦ [f])(x0) = lim (g ◦ f)(xn) = lim (g(f(xn)) n→∞ n→∞ = T ([g]) T ([f])(x0) = T ([g]) ◦ T ([f]) (x0). Note finally that

T ([idQ\X ]) = idX .  Following the definition of coarsely equivalent coarse spaces given in [14, page 25] we have: Corollary 3. Given a metric Hilbert cube (Q, d), two compact spaces X, Y ∈ OB(Z) are homeomorphic if and only if the coarse spaces (Q\X, E0) and (Q\Y,E0) are coarsely equivalent. Remark 3.1. Note that in our results we fix a metric d in the Hilbert cube. In fact if we consider another equivalent metric d andwehavetwoZ-setsX ⊂ (Q, d) and X ⊂ (Q, d), then X and X are homeomorphic if and only if the coarse spaces Q \ X and Q \ X with the corresponding C0-coarse structures are coarsely equivalent. On the other hand the concept of Z-set is independent of the choice of metric. By a uniformly continuous homeomorphism between two metric spaces we mean a homeomorphism f such that both f and f −1 are uniformly continuous. Observe also that a homeomorphism f :(Q \ X, d) −→ (Q \ Y,d)isaC0-coarse map if and only if the map f, but perhaps not f −1, is uniformly continuous. Proposition 4. Let X, Y ∈ OB(Z) be two compact spaces in a metric Hilbert cube (Q, d). a) If f :(Q \ X, d) −→ (Q \ Y,d) is a uniformly continuous homeomorphism, then [f] is a C0-coarse equivalence. Conversely, if [α]:(Q \ X, E0) −→ (Q \ Y,E0) is a coarse equivalence, then there is a uniformly continuous homeomorphism f :(Q \ X, d) −→ (Q \ Y,d) with f ∈ [α]. b) For any hereditary shape equivalence h : X −→ Y ,thereexistsaC0-coarse homeomorphism f :(Q \ X, E0) −→ (Q \ Y,E0) such that T ([f]) = h.

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Conversely, if f :(Q\X, E0) −→ (Q\Y,E0) is a C0-coarse homeomorphism, then T ([f]) : X −→ Y is a hereditary shape equivalence. c) For any C0-coarse map α :(Q \ X, E0) −→ (Q \ Y,E0) there is a proper and uniformly continuous map f :(Q \ X, d) −→ (Q \ Y,d) such that α and f are C0-close. Proof. Note that the proof of c) was really done in the demonstration of Theorem 2. Observe also that b) was in fact stated in [11]. To prove a) we only need to apply the Extension Homeomorphism Theorem ([4]) because if [α]:(Q \ X, E0) −→ (Q \ Y,E0) is an equivalence in C0(Z), then T ([α]) : X −→ Y is a surjective homeomorphism. Hence, there is a surjective home- omorphism of pairs F :(Q, X) −→ (Q, Y )withF |X = T ([α]) and, consequently, −1 −1 [F |Q\X ]=[α]and[F |Q\Y ]=[α] . The converse is obvious.  In order to strengthen our construction above with the development of coarse geometry we have the following result (for the concepts of Higson-Roe compactifi- cation and Higson-Roe corona, see [14]). Proposition 5. Given a metric Hilbert cube (Q, d) and a compact space X ∈ OB(Z), the Higson-Roe compactification of the coarse space (Q \ X, E0) is precisely Q and its Higson-Roe corona is X. Proof. Following [14], page 29, it is sufficient to prove that the C∗-algebra formed by the Higson functions on the coarse space (Q\X, E0) is *-isomorphic to C(Q) (the C∗-algebra of all complex valued continuous functions on the Hilbert cube). To do so, recall that a Higson function on (Q \ X, E0) is a bounded continuous function f : Q \ X −→ C such that the map df :(Q \ X) × (Q \ X) −→ C (x, y) → f(y) − f(x),

when restricted to any controlled set E ∈E0, vanishes at infinity. Note that if g ∈ C(Q), then g |Q\X is a Higson function. Conversely, consider a Higson func- tion f : Q \ X −→ C and fix x ∈ X. Take a sequence {αn}⊂Q \ X converging to x. Consider the C0-controlled set E = {(αn,αn+1): n ∈ N}. Using analo- gous arguments to those in Theorem 2 we obtain that F (x) = lim f(αn) defines n→∞ a continuous extension of f to Q. Sowehaveprovedthatf is a Higson func- tion on (Q \ X, E0) if and only if f :(Q \ X, d) −→ C is uniformly continuous (when considering the usual metric on C).Since(Q, d) is the metric completion of (Q\X, d), a continuous complex function f on Q\X is a Higson function if and only if f is continuously extendable to Q. This fact, jointly with the density of Q \ X in Q, provides a *-isomorphism between the C∗-algebra of the Higson functions on the coarse space (Q \ X, E0)andC(Q). 

In fact more can be said. If we denote by E1 the continuously coarse structure induced by the compactification Q in Q \ X, see [14], we have

Proposition 6. E0 = E1.

Proof. First, take E ∈E0. We are going to prove that (see [14], Theorem 2.27.a) A = ClQ×Q(E) ∩ (Q × Q) \ (Q \ X) × (Q \ X) ⊂ ∆X .

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Considerapoint(x0,y0) ∈ A and a sequence {(xn,yn)}n∈N ⊂ E such that

lim (xn,yn)=(x0,y0). n→∞

If we assume x0 ∈ X and x0 = y0,wehaved(x0,y0)=δ>0. Define ε = δ/2and let K be a compact subset of Q \ X.SinceQ \ K is an open neighborhood of X, there exists a natural number n1 such that xn ∈ Q \ K for all n ≥ n1. Due to the convergence of {xn}n∈N to x0 and {yn}n∈N to y0,thereexistsn2 ∈ N, n2 ≥ n1, such that ∈ \ × d(xn2 ,yn2 ) >δ/2and(xn2 ,yn2 ) E (K K).

Therefore E/∈E0, a contradiction. On the other hand, take E ∈E1 and assume that E/∈E0,thatistosay, there exists ε>0 such that for each compact Kn = Q − B(X, 1/n) ⊂ Q \ X, n ∈ N, it is possible to choose (xn,yn) ∈ E \ (Kn × Kn) such that d(xn,yn) ≥ ε. From the compactness of Q × Q, the sequence {(xn,yn)}n∈N has an accumulation point (x0,y0). Necessarily this accumulation point belongs to the closure of E in Q × Q and, by the construction of the sequence {(xn,yn)}n∈N, x0 ∈ X or y0 ∈ X. Therefore (x0,y0) ∈/ (Q\X)×(Q\X). As d(xn,yn) ≥ ε for all n ∈ N, d(x0,y0) ≥ ε. Hence x0 = y0 and (x0,y0) ∈/ ∆X . This contradicts the fact that E ∈E1 (see [14], Theorem 2.27.a). 

3. Relating invariants of Z-sets and their complements Now we can reinterpret some topological invariants of a Z-set X in terms of the coarse space (Q \ X, E0). In particular Corollary 7. Let X be a Z-set in the Hilbert cube Q. X is finite dimensional if and only if the asymptotic dimension of (Q \ X, E0) is finite. Moreover,

dim(X)=asdim(Q \ X, E0) − 1.

Proof. Since X is the Higson-Roe corona of (Q \ X, E0), the result follows from [7, Theorem 2.5.7 page 32].  Using any of the equivalent definitions of the asymptotic dimension of a coarse structure ([7, 14]), we have descriptions, from the outside, of the dimension of a Z-set in the Hilbert cube. That way we answer Question 1 in [1]. One of the many ways to describe the topological dimension of a compact metric space is that of using extensions of maps to spheres (see [10, chapter VI]). Part a) of the next result is an analog for the asymptotic dimension of (Q \ X, E0). We use this to prove part b), that is related to a result in [11], stated there without proof. Proposition 8. Consider a metric Hilbert cube (Q, d) and let Sn be any Z-copy of the n-sphere. Suppose that X ⊂ (Q, d) is a compact Z-set. Then the inequality asdim(Q \ X, E0) ≤ n +1,ordim(X) ≤ n, is equivalent to any of the following facts: a) Any uniformly continuous and proper map f : A −→ (Q \ Sn,d) is extend- able to a uniformly continuous proper map F :(Q \ X, d) −→ (Q \ Sn,d), where A is any closed subset of (Q \ X, d). ◦ b) Any uniformly continuous and proper map f : A −→ (In+1,d) is extendable ◦ to a uniformly continuous proper map F :(Q \ X, d) −→ (In+1,d),where

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◦ In+1 denotes the combinatorial interior of the (n+1)-cube, d is any metric inducing the usual topology in the cube In+1 and A is any closed subset of (Q \ X, d).

Proof. To prove a) suppose asdim(Q \ X, E0) ≤ n + 1. Using Corollary 7, it is Y equivalent to dim X ≤ n. For any Z-set Y ,wedenotebyH : Q × IQ −→ Q, Y ≡ Y ∩ ∅ where IQ =[0, diam(Q)], a homotopy such that H0 idQ and Ht (Q) Y = for any t ∈ IQ \{0} (see [3] or [4]). Fix A ⊂ Q \ X a closed subset and a uniformly continuous and proper map f : A −→ (Q \ Sn,d). Define the trace of A as Tr(A)= Q A ∩ X.Sincef is a uniformly continuous map, there is a continuous extension n f : A ∪ Tr(A) −→ Q of f with f(Tr(A)) ⊂ S due to properness. Now f |Tr(A) is extendable to a continuous map fˆ : X −→ Sn (see [10, Theorem VI.4]). Clearly the function g : X ∪ A −→ Q fˆ(y)ify ∈ X y → g(y)= f(y)ify ∈ A is continuous. Since Q is an AR, there is a continuous extension G : Q −→ Q of g. Finally consider the function F : Q −→ Q n x → F (x)=HS (G(x),d(x, X ∪ A)). n By the construction F (Q \ X) ⊂ Q \ S and it is easy to see that F |Q\X is an extension of f that satisfies all the required conditions. To prove the converse take C ⊂ X a closed subset and a continuous function n X X h : C −→ S . Consider a homotopy H as above. A = H (C × IQ) \ C is a closed subset of Q \ X whose trace is simply C.Letf :(A, d) −→ (Q \ Sn,d) be defined by n f(x)=HS (h˜(x),d(x, C)) where h˜ : Q −→ Q is any continuous extension of h. f is proper and uniformly continuous. So there is a proper and uniformly continuous extension n F :(Q \ X, d) −→ (Q \ S ,d); therefore F is a C0-coarse map. Using Theorem 2 we obtain that T ([F ]) : X −→ Sn is a continuous extension of h.Consequently dim X ≤ n and now, from Corollary 7, asdim(Q \ X, E0) ≤ n +1. To prove b) suppose that asdim(Q \ X, E0) ≤ n + 1. Consider a uniformly ◦ continuous and proper map f : A −→ (In+1,d), with A ⊂ Q \ X aclosedset.By compactness, we can suppose that In+1 is a closed Z-set in Q and that the metric d is inherited from Q. Note that the combinatorial boundary Sn of In+1 is a Z-set in Q. Therefore f(A) ⊂ Q \ Sn. Then, by a), there is a proper uniformly continuous extension f˜ : Q \ X −→ Q \ Sn. Take a retraction r : Q −→ In+1. Define n n R : Q −→ In+1 by R(x)=HS (r(x),d(x, In+1)), where HS is a homotopy realizing the fact that the combinatorial boundary is a Z-set in In+1.Observethat ◦ ◦ R(Q \ Sn) ⊂ In+1.Consequently,R ◦ f˜ : Q \ X −→ In+1 is a uniformly continuous proper extension of the initial map f. To prove the converse take C ⊂ X tobeaclosedsubsetandleth : C −→ Sn be a continuous function. Consider Sn as the combinatorial boundary of In+1. Since In+1 is an AR space there is a continuous extension h˜ : Q −→ In+1.Let n G : Q −→ In+1 be defined by G(x)=HS (h˜(x),d(x, C)). Note that G(Q \ C) ⊂

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◦ n+1 X I .ConsiderA = H (C × IQ) \ C. A is closed in Q \ X.Bythehypothesis, G |A can be extended to a proper uniformly continuous function G˜ : Q \ X −→ ◦ In+1. So there is a continuous extension to the corresponding metric completions n+1 n G : Q −→ I with G(X) ⊂ S .Byconstruction,G |X is a continuous extension of h.Consequentlydim(X) ≤ n; hence asdim(Q \ X, E0) ≤ n +1.  From the isomorphism of categories in Theorem 2 another of the topological invariants of a metric compactum that can be clearly described from the outside is the algebraic type of the group of autohomeomorphisms. In [14, page 112] the quasi-isometry group of a coarse space Y ,Qis(Y ), is introduced as the group of closeness classes of coarse equivalences in Y .ForeachZ-setX in the Hilbert cube (Q, d), where d is any fixed metric, denote by Hu(Q \ X) the group of uniform autohomeomorphisms of (Q \ X, d). Let Hid(Q \ X) be the normal subgroup of Hu(Q \ X) defined by

Hid(Q \ X)={f ∈ Hu(Q \ X):T ([f]) = idX }.

Note that Hid(Q\X) is, up to the obvious identification, the subgroup of the group H(Q) (of all autohomeomorphisms of the Hilbert cube) leaving fixed all points in X. Using the Homeomorphism Extension Theorem for part b), one can prove (we leave it to the reader): Corollary 9. The group of autohomeomorphisms H(X) of a compactum X ∈Z is isomorphic to the following groups:

a) Qis(Q \ X),whereweconsiderQ \ X as an object in C0(Z); H (Q \ X) b) u . Hid(Q \ X) Using again the isomorphism T constructed in Theorem 2 one can prove the following: Proposition 10. Let (Q, d) be a fixed metric Hilbert cube. The homotopy category of compact Z-sets in Q (and the homotopy category of compact metrizable spaces) is isomorphic to the proper uniformly continuous homotopy category of their comple- ments with the inherited metrics. This isomorphism transforms any compact Z-set into its complement.

Proof. Denote by C1 the category whose objects are the complements, in Q,of compact Z-sets and the morphisms are the uniform and proper homotopy classes of uniformly continuous and proper functions. Let C2 be the category whose ob- jects are the compact Z-sets in Q and the morphisms are the classes of homotopic continuous functions. We assert that there is an isomorphism of categories

[T ]:C1 −→ C 2 whose action on objects is defined by [T ](Q \ X)=X andonmorphismsby

[T ]([f]) = [T (f)]H

where T is the isomorphism of categories constructed in Theorem 2 and [T (f)]H is the homotopy class of the map T (f).

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[T ] is well-defined: Given two homotopic proper and uniformly continuous func- tions f,g : Q \ X −→ Q \ Y there exists a proper and uniformly continuous homotopy H :(Q \ X) × I −→ Q \ Y from f ≡ H(·, 0) to g ≡ H(·, 1). As (Q \ X) × I =(Q × I) \ (X × I), (Q × I)is homeomorphic to Q and X × I is a compact Z-set in the Hilbert cube (Q × I), it is possible to extend H to another homotopy Hˆ : Q × I −→ Q such that Hˆ |X×I : X × I −→ Y. Since Hˆ (·, 0) and Hˆ (·, 1) are continuous extensions to Q of f and g respectively, nec- essarily the homotopy Hˆ |X×I connects T (f)withT (g). In this way, the definition of the functor [T ] on morphisms is correct. [T ]isfaithful: Suppose that [T ]([f1]) = [T ]([f2]) for some maps f1,f2 : Q\X −→ Q\Y . As the maps T (f1),T(f2):X −→ Y are homotopic, there exists a homotopy G : X × I −→ Y such that G(·, 0) ≡ T (f1)andG(·, 1) ≡ T (f2). Recall that if we paste the functions fi and T (fi)(i ∈{1, 2}), they give rise to continuous functions ˆ ˆ f1, f2 : Q −→ Q. Consider the closed subset of Q × I, C =(X × I) ∪ (Q ×{0}) ∪ (Q ×{1}), and define the continuous function H˜ : C −→ Q ⎧ ˆ ⎨⎪f1(x)ift =0 (x, t) → H˜ (x, t)= fˆ (x)ift =1 ⎩⎪ 2 G(x, t)ifx ∈ X. Since Q is an AR space, H˜ extends to a continuous map Hˆ : Q × I −→ Q.Denote by HY , as in previous results, a homotopy related to the compact Z-set Y and define the homotopy H : Q × I −→ Q (x, t) → H(x, t)=HY (H˜ (x, t),d((x, t),C)).

Clearly, H(·, 0) ≡ fˆ1, H(·, 1) ≡ fˆ2 and H(x, t)=G(x, t) for all x ∈ X and t ∈ I. Due to the fact that H−1(Y )=X × I, the restriction

H |(Q\X)×I :(Q \ X) × I −→ Q \ Y

is a proper and uniformly continuous homotopy between f1 and f2. [T ]isfull: It follows straightforwardly from the fullness of the functor T (see Theorem 2).  In the proposition above the phrase proper uniformly continuous homotopy cat- egory means that the homotopy should also be proper and uniformly continuous if one considers the maximum metric when crossing by a closed bounded real interval. In the remainder of the paper we are going to concentrate on the task of de- scribing the ANR-property of a compact Z-set by means of its complement. The

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main reason is that the ANR-property is a necessary one for being a manifold. The following results could be considered a modest contribution to the second question in [1]. We think they are of their own interest because the ANR-property is also important for other problems in topology. The concepts of Absolute Neighborhood Retract (ANR) and Absolute Neighbor- hood Extensor (ANE) are equivalent for metrizable spaces [2, 9]. We are going to define our corresponding candidates from the outside. First of all fix any metric d in Q inducing its topology. We are going to consider (Q\X, d |(Q\X)×(Q\X)) for any compact Z-set X in Q. From now on we omit the reference to the induced metrics but understanding that when we refer to uniform properties we mean related to those metrics. So, from now on, f : Q\X −→ Q\Y is a uniform embedding if f is a topological embedding such that both f and f −1, in the corresponding domain, are uniformly continuous with the corresponding relative metrics. By a uniform neigh- borhood of a subset we mean a neighborhood which contains an ε-neighborhood of the subset. Let us consider the following.

Definition 1. Q\X is a C0-ANR space if for every uniform embedding f : Q\X −→ Q\Y , as closed subset, there exists a uniform neighborhood U of f(Q\X)inQ\Y and a retraction r : U −→ f(Q \ X) that is uniformly continuous and proper.

Definition 2. Q \ X is a C0-ANE space if every uniformly continuous and proper map from a closed subset A˜ of Q \ Y to Q \ X admits a uniformly continuous and proper extension to a uniform neighborhood of A˜ in Q \ Y . The following result will be useful. Proposition 11. If X and Y are compact Z-sets in (Q, d),thenQ \ X admits a uniform embedding, as a closed subset, into Q\Y if and only if X can be topologically embedded into Y . Proof. If f : Q \ X −→ Q \ Y is a closed uniform embedding and F : X −→ Y is the natural map induced by f as in Theorem 2, then F is clearly a topological embedding. To prove the remaining part we can suppose that the topological embedding of X into Y is just an inclusion. So consider X ⊂ Y. Finally consider Q ×{0}⊂Q × I.NotethatQ ×{0} is an isometric copy of (Q, d) if we consider the maximum metric in Q × I which is again a Hilbert cube. Let µ : Q −→ [0, 1] be a continuous function such that µ−1(0) = X. Clearly h : Q −→ Q × I q → (q, µ(q)) is a topological embedding such that h(Q) ∩ Q ×{0} = X ×{0}.Moreover, h(Q \ X) ⊂ Q × (0, 1] ⊂ Q × I \ Y ×{0}. Furthermore h(Q \ X)isclosedinQ × I \ Y ×{0}. Using now the Homeomorphism Extension Theorem and the fact that Y ×{0} is a Z-set in Q × I one easily obtains that (Q \ X, d) can be uniformly embedded as a closed subset in (Q \ Y,d). 

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From now on we are going to denote by Q any Z-copy of the Hilbert cube inside (Q, d). Now we have

Proposition 12. Q \ X is a C0-ANR space if and only if for every uniform em- bedding of Q \ X in Q \Q, as a closed subset, there exists a uniformly continuous and proper retraction r : U −→ Q \ X,whereU is a uniform neighborhood of Q \ X in Q \Q. Proof. From the previous proposition, Q \ X admits a uniform embedding j : Q \ X −→ Q \Q, as a closed subset, into Q \Q. Suppose that Q \ X is a C0-ANR space. From the hypothesis, there exists a uniform neighborhood U of j(Q \ X)inQ \Qand a retraction r : U −→ j(Q \ X) that is uniformly continuous and proper. Conversely, let f : Q \ X −→ Q \ Y be a uniform embedding, as a closed subset. Take i : Q \ Y −→ Q \Qto be a closed uniform embedding. The composition i ◦ f : Q \ X −→ Q \Q is a uniform embedding of Q \ X into Q \Qas a closed subset. So there exists a uniformly continuous and proper retraction r : U −→ (i ◦ f)(Q \ X), where U is a uniform neighborhood of (i ◦ f)(Q \ X)inQ \Q.Sincei is a uniformly continuous map, there exists a uniform and closed neighborhood of the subset f(Q \ X)in Q \ Y , W , such that i(W ) ⊂ U. In this manner, the function r : W −→ f(Q \ X) z → (i−1 ◦ r ◦ i)(z) is uniformly continuous and a proper retraction.  We will also establish the following equivalence.

Proposition 13. Q\X is a C0-ANR space if and only if Q\X is a C0-ANE space.

Proof. Assume that Q\X is a C0-ANR space and let f : A˜ −→ Q\X be a uniformly continuous and proper map from a closed subset A˜ of Q \ Y .Sincef is a proper map, Q is an AE space and X is a Z-set, we can extend it to a continuous function F : Q −→ Q such that F −1(X) ⊂ Y . Consider a uniform embedding j : Q \ X −→ Q \Q

as a closed subset. As Q\X is a C0-ANR, there exists a uniform neighborhood U of j(Q \ X)inQ \Qand a retraction r : U −→ j(Q \ X) that is uniformly continuous and proper. Moreover, as in the proof of Proposition 11, it is possible to extend j to a continuous function ˜j : Q −→ Q such that ˜j |X : X −→ ˜j(X) ⊂Qis a homeomorphism. Since ˜j ◦ F : Q −→ Q is a uniformly continuous map, there exists a uniform and closed neighborhood of A˜ in Q \ Y , W , such that (˜j ◦ F )(W ) ⊂ U. In this way, the function f˜ : W −→ Q \ X defined by ˜ −1 f ≡ ˜j ◦ r ◦ ˜j ◦ F |W is uniformly continuous and is a proper extension of f. Now suppose that Q \ X is a C0-ANE space and let i : Q \ X −→ Q \Q

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be a uniform embedding, as a closed subset, of Q \ X in Q \Q. Clearly i−1 : i(Q \ X) −→ Q \ X is a uniformly continuous and proper map from a closed subset i(Q \ X)ofQ \Q to Q \ X. Therefore i−1 admits a uniformly continuous and proper extension to a uniform neighborhood U of i(Q \ X)inQ \Q: s : U −→ Q \ X. Finally, r ≡ i ◦ s : U −→ i(Q \ X) is uniformly continuous and a proper retraction. 

The following result, which is inspired by [11], is our characterization, from the outside, of the ANR-property of compacta. We prefer to use the extension version in the outside instead of the retraction version used in a related analogous result of [11]. Proposition 14. Let X be a compact metric space embedded in the Hilbert cube (Q, d) as a Z-set. Q \ X is a C0-ANE space if and only if X is an ANR space. Proof. Let X be a compact metric space embedded in the Hilbert cube (Q, d)as a Z-set, such that Q \ X is a C0-ANE space. Take a closed subset A of another compact metric space Y and a continuous function f : A −→ X.Fromahomotopy HY : Q × I −→ Q associated to the Z-set Y , we construct a closed subset of Q \ Y defined by A˜ = HY (A × I) \ A.AsQ is an AR space there exists a continuous extension f˜ of f, f˜ : A˜ ∪ A −→ Q. The function ˆ ˜ ∪ −→ f : A A Q z → fˆ(z)=HX f˜(z),d(z,A)

is continuous between compacta. Since fˆ−1(X)=A, the restriction ˆ | ˜ −→ \ f A˜: A Q X is a uniformly continuous and proper map. Hence, by hypothesis, it is possible to extend it, by means of another uniformly continuous and proper function h,toa uniform closed neighborhood of A˜ in Q \ Y , h : B(A,˜ ε) ∩ (Q \ Y ) −→ Q \ X.

The properness of h allows us to extend it continuously from ClQ(B(A,˜ ε)∩ (Q\Y )) = B(A,˜ ε)toQ, h˜ : B(A,˜ ε) −→ Q. Since B(A, ε) ∩ Y ⊂ B(A,˜ ε) ∩ Y and, necessarily, h˜(B(A,˜ ε) ∩ Y ) ⊂ X, ˜ | ∩ −→ h B(A,ε)∩Y : B(A, ε) Y X is a continuous map which extends f to a closed neighborhood of A in Y . Conversely, let X be an ANR compact metric space embedded in the Hilbert cube (Q, d) as a Z-set. Consider a closed subset A˜ of Q \ Y ,whereY is another compact metric space embedded in the Hilbert cube (Q, d) as a Z-set, and a uniformly continuous and proper map f : A˜ −→ Q \ X.

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a) If Tr(A˜)=∅,thenA˜ is compact and the function

g : A˜ ∪ Y −→ Q f(z) , if z ∈ A˜ z → g(z)= x0 , if z ∈ Y

for a fixed point x0 ∈ X, is continuous. From a continuous extension F : Q −→ Q of g define the map ˜ F : Q −→ Q x → F˜(x)=HX F (x),d(x, A˜ ∪ Y ) .

Since F˜ is a uniformly continuous proper map and F˜−1(X)=Y ,there- striction ˜ F |Q\Y : Q \ Y −→ Q \ X is also uniformly continuous and a proper map which extends f. b) If Tr(A˜)=ClQ(A˜) ∩ Y = A = ∅, there exists a continuous extension of ˜ ˜ f, f : A˜ ∪ A =ClQ(A˜) −→ Q such that f(A) ⊂ X.AsX is an ANR ˜ ˆ space, the map f |A: A −→ X admits a continuous extension, f,toa closed neighborhood of A in Y .SinceA is compact, we can assume that this closed neighborhood is B(A, ε) ∩ Y , for certain ε>0, fˆ : B(A, ε) ∩ Y −→ X. From their coincidence in A, it is possible to paste the functions f˜ and fˆ. That way one obtains the continuous map f : B(A, ε) ∩ Y ∪ A˜ −→ Q f˜(x) , if x ∈ A˜ x → f(x)= fˆ(x) , if x ∈ B(A, ε) ∩ Y.

Since the compact sets A˜ ∪ A and Y \ B(A, ε) are disjoint, there exists a number δ,0<δ<ε, such that B(A,˜ δ) ∩ Y ⊂ B(A, ε). As Q is an AR space, the map f can be continuously extended to a function g : B(A, ε) ∩ Y ∪ B(A,˜ δ) −→ Q. Define ˜ g˜ : B(A, ε) ∩ Y ∪ B(A, δ) −→ Q x → g˜(x)=HX g(x),d(x, B(A, ε) ∩ Y ∪ A˜) . Sinceg ˜ is a uniformly continuous and proper map andg ˜−1(X)=B(A, ε)∩ Y , the restriction g˜ | : B(A,˜ δ) \ Y −→ Q \ X B(A,δ˜ ) \Y is also uniformly continuous and a proper map which extends f to a uniform closed neighborhood of A˜ in Q \ Y . 

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5246 E. CUCHILLO-IBA´ NEZ,˜ J. DYDAK, A. KOYAMA, AND M. A. MORON´

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Departamento Matematica´ Aplicada, E.T.S.I. Montes, Universidad Politecnica,´ 28040 Madrid, Spain E-mail address: [email protected] Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996 E-mail address: [email protected] Department of Mathematics, Shizuoka University, Shizuoka, Japan E-mail address: [email protected] Departamento Geometr´ıa y Topolog´ıa, Facultad de Cc.Matematicas,´ Universidad Complutense, 28040 Madrid, Spain E-mail address: ma [email protected]

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